Invertibility; properties of y[n], y(t)

Invertibility

A system is invertible if distinct inputs always produce distinct outputs, so that the input can be recovered uniquely from the output by an inverse system. Example: y(t) = 2x(t) is invertible (x = y/2); y(t) = x²(t) is not (x and -x give the same y).

(i) y[n] = 3x[n] - 4x[n - 1]

• Linear: yes (a weighted sum of inputs).
• Causal: yes (uses only the present and the previous input).
• Time-invariant: yes (constant coefficients).
• Stable: yes (an FIR system; |y| ≤ 7M for |x| ≤ M).
• Memory: yes (depends on x[n - 1]).
• It is also invertible: x[n] = (y[n] + 4x[n - 1])/3, though the inverse system is unstable.

(ii) y(t) = sin(2π x(t))

• Linear: no, because sin(2π(a·x₁ + b·x₂)) ≠ a·sin(2πx₁) + b·sin(2πx₂).
• Causal: yes (depends on the present input only).
• Time-invariant: yes (a delayed input gives the same delayed output).
• Stable: yes, since |y(t)| ≤ 1 for every input.
• Memory: no, it is memoryless.
• Not invertible: x(t) and x(t) + 1 give the same output.